# Cournot-Nash Equilibrium in Duopoly

This is a homework question, but resources online are exceedingly complicated, so I was hoping there was a fast, efficient way of solving the following question:

There are 2 firms in an industry, which have the following total cost functions and inverse demand functions. \begin{align*} \text{Firm 1}: &C_1 =50 Q_1\\ &P_1 = 100 \,– 0.5 (Q_1 + Q_2)\\ \text{Firm 2}: &C_2 = 24 Q_2\\ &P2 = 100 \, – 0.5 (Q_1 + Q_2) \end{align*} What is the Cournot-Nash equilibrium for this industry?

I've tried to solve this dozens of times. My idea was to find the profit equation for both, take the derivative, set equal to zero, and then solve for $Q_1$ and $Q_2$.

Doing this, I get: $$Q_1 = -5Q_2 + 500\\ Q_2 = -5Q_1 + 760$$ Plug $Q_2$ equation in Firm 1's equation and solve, but I keep getting that $Q_1$ should equal $137.5$, which is not the correct answer.

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There is a standard way of solving for $Q_1$ and $Q_2$.

1. Determine the profit functions.

2. Determine the best response function for the firms.

3. Substitute $Q_1$ or $Q_2$ in the other profit function and solve.

All these steps are already mentioned, so you know what to do. Below you can search for your mistake.

The profit function for firm 1 equals $\Pi_1= P_1Q_1-C_1=Q_1 \cdot (100-0.5(Q_1+Q_2)) - 50Q_1$

The profit function for firm 2 equals $\Pi_2=P_2Q_2-C_2=Q_2 \cdot (100-0.5(Q_1+Q_2)) - 24Q_2$

The best response function can be determined by deriving the profit function of firm 1 w.r.t. $Q_1$ and for firm 2 w.r.t. $Q_2$ and set them equal to zero

$$\frac{\partial \Pi_1}{\partial Q_1}=100-Q_1-0.5Q_2-50=50-Q_1-0.5Q_2=0$$

$$\implies Q_1=50-0.5Q_2$$

$$\frac{\partial \Pi_2}{\partial Q_2}=100-Q_2-0.5Q_1-24=76-Q_2-0.5Q_1=0$$

Now we can make the substitution

$$76-Q_2-0.5 \cdot (50-0.5Q_2)=0$$

$$\implies 51-Q_2+0.25Q_2=0 \implies 0.75Q_2=51$$

And thus we find $Q_2=68$ and can solve easily for $Q_1$ $$Q_2=68 \ \text{and} \ Q_1=50-0.5 \cdot 68=16$$

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This is perfect. Thank you. –  Parseltongue Mar 11 '13 at 23:44
Your welcome :) –  Bob Mar 11 '13 at 23:53

To solve with Mathematica, see example in Simple Cournot example in Mathematica. Your best response functions $Q1 = -5Q2 + 500$, $Q2 = -5Q1 + 760$ are incorrect. For $P=a-b(Q_1+Q_2)$ and $C_1(Q_1)=c\cdot Q_1$, the profit function is $\Pi(Q_1,Q_2)=(a-bQ_2-c_1) Q_1-bQ_1^2$, so the best response is $Q_1(Q_2)=\frac{a-bQ_2-c_1}{2b}$. In your case, $a=100$, $b=0.5$, $c_1=50$, which results in $Q_1(Q_2)=\frac{100-50}{2\cdot0.5}-\frac{1}{2}Q_2$.

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I don't have mathematica, so I can't test this. –  Parseltongue Mar 11 '13 at 19:19